Never run away from the Time Police!

Well, that depends on which time travel rules we're following. If it's Star Trek time travel rules, then no, because by creating a new timeline you break the timeline you originally came from and send it in a different direction, and your past self from the new timeline doesn't necessarily need to fulfil a predestination paradox and come back and do shit—so there's no responsibility.

Terminator time travel rules, however, see things a little differently. There might be several versions of the same timeline, but in the end, whatever happens, the timeline corrects itself so certain events always wind up occurring. Judgment Day gets delayed, but still takes place, and John Connor sends back Kyle Reese and later a modified Terminator no matter what. This, however, raises the question of whether anyone following Terminator rules can be held accountable for any of their actions, as, seemingly, the timeline will correct itself to make the same rough series of events occur despite efforts to the contrary. It's a bit hard to decide on this one, as it deals with free will and destiny and the like.

Back To The Future rules work a little like Star Trek rules, with the exception that everything can be put back to normal if the world gets a little too muddled as a result of wacky time-traveling adventures. Given that restoring the timeline to its original state wouldn't be noticeable to anyone except the person/people who fixed it after screwing it up, nobody will know any different, and they certainly won't need to be reprimanded for their actions.

Lastly, I'll mention Primer's rules, which hold that causing the slightest discrepancy in the timeline will fuck things up hugely, and that sooner or later people will notice that bad things are happening. In those circumstances, as soon as anyone incurs a paradox of any kind, I reckon they, along with their duplicates, ought to be shot on sight, as by that point they could cause no end of mayhem simply by continuing to exist.

Jam it back in, in the dark.